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We solve the quantum version of the $A_1$ $T$-system by use of quantumnetworks. The system is interpreted as a particular set of mutations of asuitable (infinite-rank) quantum cluster algebra, and Laurent positivityfollows from our solution. As an application we re-derive the correspondingquantum network solution to the quantum $A_1$ $Q$-system and generalize it tothe fully non-commutative case. We give the relation between the quantum$T$-system and the quantum lattice Liouville equation, which is the quantized$Y$-system.

The solution of the quantum $A_1$ T-system for arbitrary boundary